Question

Every linear transformation from R^n to R^m is a matrix transformation. Choose the correct answer below.

A. True. Every linear transformation can be represented by a matrix.
B. False. Some linear transformations have no matrix representation.
C. True. Only transformations from R^2 to R^2 have matrix representations.
D. False. Matrix transformations are limited to certain vector spaces.

Answer 1

True, every linear transformation from ℝ^n to ℝ^m can be represented by a matrix. This holds because these transformations preserve vector addition and scalar multiplication, and can be expressed by their action on basis vectors. The correct option is A.

Step-by-step explanation:

The question revolves around the concept of linear transformations in vector spaces, particularly those from ℝ^n to ℝ^m. When dealing with vector spaces, any linear transformation can indeed be represented by a matrix. This is because a matrix can act on vectors by multiplying them, thus effecting a transformation.

Specifically, the action of a linear transformation on the standard basis vectors of ℝ^n determines the columns of the corresponding transformation matrix, which can then be applied to any vector in ℝ^n to yield a transformed vector in ℝ^m.

In light of this, the correct answer to the student's question is A. True. Every linear transformation can be represented by a matrix. This is a fundamental principle in linear algebra, and it holds on the basis that linear transformations satisfy two main properties: they must preserve vector addition and scalar multiplication. Hence, they can be fully described by the effects based on the source space, which in turn determines a unique matrix.